Stability of Rigid Body Rotations

Consider a general rigid body with $I^{(1)}\leq I^{(2)}\leq I^{(3)}$, rotating about one of the principal axes. If small rotations about the other principal axes are introduced, will the rotation be stable? (What happens if you toss a book in the air?)

Lets start with rotations about the first principal axis with the smallest moment of inertia.

$$\vec{\omega}=\omega_1\hat{e}_{1}+\epsilon\hat{e}_{2}+\eta\hat{e}_{3}$$

We assume $\epsilon$ and $\eta$ are very small. Applying the Euler Equations for no applied torque, we get:

$$I^{(k)}\dot{\omega}_k + I^{(j)}\omega_j\omega_i\epsilon_{ijk} = 0$$

$$I^{(1)}\dot{\omega}_1 + I^{(3)}\omega_3\omega_2 - I^{(2)}\omega_2\omega_3= 0$$

$$I^{(1)}\dot{\omega}_1 + \left(I^{(3)}- I^{(2)}\right)\epsilon\eta = 0$$

$$I^{(2)}\dot{\epsilon} + \left(I^{(1)}-I^{(3)}\right)\omega_1\eta = 0$$

$$I^{(3)}\dot{\eta} + \left(I^{(2)}-I^{(1)}\right)\omega_1\epsilon = 0$$

We may neglect terms of order of the perturbation squared so the three coupled equations become.

$$I^{(1)}\dot{\omega}_1 = 0$$

$$I^{(2)}\dot{\epsilon} + \left(I^{(1)}-I^{(3)}\right)\omega_1\eta = 0$$

$$I^{(3)}\dot{\eta} + \left(I^{(2)}-I^{(1)}\right)\omega_1\epsilon = 0$$

$$\omega_1=const.$$

$$\dot{\epsilon} = {I^{(3)}-I^{(1)}\over I^{(2)}}\omega_1\eta = 0$$

$$\dot{\eta} = {I^{(1)}-I^{(2)}\over I^{(3)}}\omega_1\epsilon = 0$$

Note that the constants in the equations are different so we must differentiate one equation again to get a solution.

$$\ddot{\epsilon} = {I^{(3)}-I^{(1)}\over I^{(2)}}\omega_1\dot{\eta}$$

$$\ddot{\epsilon} = {\left(I^{(3)}-I^{(1)}\right)\left(I^{(1)}-I^{(2)}\right)\over I^{(2)} I^{(3)}}\omega_1^2\epsilon$$

If ${\left(I^{(3)}-I^{(1)}\right)\left(I^{(1)}-I^{(2)}\right)\over I^{(2)} I^{(3)}}$ is negative, the solutions are oscillatory. If it is positive, they can grow exponentially. The constant is negative for axis 1. (Note that the $\eta$ equation has the same constant.)

The equation can actually be applied to any of the three axes. For axis 2, the constant is ${\left(I^{(1)}-I^{(2)}\right)\left(I^{(2)}-I^{(3)}\right)\over I^{(3)} I^{(1)}}$ which is positive, and the oscillations are unstable. For axis 3, the constant is ${\left(I^{(2)}-I^{(3)}\right)\left(I^{(3)}-I^{(1)}\right)\over I^{(1)} I^{(2)}}$ which is negative and the oscillations are stable.

So rotations about the axes with the lowest and highest moment are stable while oscillations about the axis with the intermediate moment are unstable. Try it with a book.